What does it mean to have no proper non-trivial subgroup

For example, in $\{0,1,2,3\}$ (cyclic group of order $4$) the elements $\{0,2\}$ make a subgroup. This is a nontrivial subgroup, and it is not the entire group, so it is a proper subgroup.

The point is that a subgroup is also a subset. Subsets can be proper, or improper (i.e. equal to the big set). Proper subgroups are proper subsets.

A subgroup N of a group $G$ is said to be proper if $N\neq G$ and to be non-trivial if $N\neq \{e\}$, where $e$ is the identity of $G$.

For example $N=\{0,2\}$ is a proper subgroup of $(\Bbb Z/4\Bbb Z,+)$, isomorphic to $\Bbb Z/2\Bbb Z$.